There are versions of the KHA for any stability condition of \(Q\) . Davison–Meinhardt proved the PBW theorem for a generic stability condition on an arbitrary pair \((Q,W)\) ; when the stability condition is zero, \(Q\)  has to be symmetric. There are also \(T\) -equivariant versions of the KHA for a torus \(T\subset \left(\mathbb {C}^*\right)^E\)  which fixes \(W\) . It is interesting to prove versions of Theorems REF  and REF  in these more general cases. In the case of tripled quivers and certain equivariant actions, the algebra morphism (REF ) is an inclusion [1]}, see also [2]} for the analogous statement in cohomology, and thus an equivariant version of Theorem REF  would imply a PBW theorem for the full preprojective KHA.
